Maximum likelihood estimation of time and frequency offset for OFDM systems

ABSTRACT

A receiver in an OFDM system may include a joint maximum likelihood (ML) estimator that estimates both time offset and frequency offset. The estimator may use samples in an observation window to estimate the time offset and frequency offset.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application Ser.No. 60/598,585, filed on Aug. 2, 2004.

BACKGROUND

Wireless systems may use an Orthogonal Frequency Division Multiplexing(OFDM) transmission scheme. In an OFDM system, a data stream is splitinto multiple substreams, each of which is sent over a subcarrierfrequency. Because data is carried over multiple carrier frequencies,OFDM systems are referred to as “multicarrier” systems as opposed tosingle carrier systems in which data is transmitted on one carrierfrequency.

An advantage of OFDM systems over single carrier systems is theirability to efficiently transmit data over frequency selective channelsby employing a fast Fourier transform (FFT) algorithm instead of thecomplex equalizers typically used by single carrier receivers. Thisfeature enables OFDM receivers to use a relatively simple channelequalization method, which is essentially a one-tap multiplier for eachtone.

Despite these advantages, OFDM systems may be more sensitive to timeoffset than single carrier systems. Also, demodulation of a signal withan offset in the carrier frequency can cause a high bit error rate andmay degrade the performance of a symbol synchronizer in an OFDM system.

SUMMARY

A receiver in an Orthogonal Frequency Division Multiplexing (OFDM)system may include a joint maximum likelihood (ML) estimator thatestimates both time offset and frequency offset. The estimator may usesamples in an observation window to estimate the time offset andfrequency offset. The size of the observation window may be selected tocorrespond to a desired performance level of the estimator, with largerwindow sizes providing better accuracy.

The estimator may include a receiver to receive a number of symbols,each symbol including a body samples and cyclic prefix samples, a framecircuit to observe a plurality of samples in the window, and acalculator to calculate a correction value based on a number ofcorrelated cyclic prefix samples and body samples in the window and tocalculate an estimated time offset value using samples in the window andthe correction value.

The estimator may calculate the estimated time offset value {circumflexover (θ)} by solving the equation:

${\hat{\theta} = {\underset{0 \leq m \leq {N_{t} - 1}}{\arg\;\max}\left\{ {{{\gamma(m)}} - {\rho\;{\Phi(m)}} - {\Psi(m)}} \right\}}},$

where

$\begin{matrix}{{{\gamma(m)} = {\sum\limits_{n \in {A_{m}\bigcap I}}\;{{y\lbrack n\rbrack}{y^{*}\left\lbrack {n + N} \right\rbrack}}}},} \\{{{\Phi(m)} = {\frac{1}{2}{\sum\limits_{n \in {A_{m}\bigcap I}}\left( {{{y\lbrack n\rbrack}}^{2} + {{y\left\lbrack {n + N} \right\rbrack}}^{2}} \right)}}},} \\{{{\Psi(m)} = {{{A_{m}\bigcap I}}\frac{2\rho}{\left( {\sigma_{x}^{2} + \sigma_{z}^{2}} \right)\left( {1 - \rho^{2}} \right)}{\log\left( {1 - \rho^{2}} \right)}}},} \\{{\rho = \frac{\sigma_{x}^{2}}{\sigma_{x}^{2} + \sigma_{z}^{2}}},} \\{{A_{m} = {\underset{i = {- \infty}}{\bigcup\limits^{\infty}}\left\{ {{m + {iN}_{t}},\ldots\mspace{14mu},{\left( {N_{g} - 1} \right) + m + {iN}_{t}}} \right\}}},}\end{matrix}$

I={0, . . . , M−N−1}

where y[n] is a received sample with index n, N is the number of samplesin the body of a symbol, N_(g) is the number of samples in the cyclicprefix of a symbol, N_(t) is the total number of samples in a symbol, Mis the number of samples in the window, σ_(x) ² is the variance of atransmit signal, σ_(z) ² is the variance of the white Gaussian noise,and Ψ(m) is the correction value.

The estimator may calculate the estimated frequency offset value{circumflex over (ε)} by solving the following equation:

$\hat{ɛ} = {{- \frac{1}{2\pi}}{{{\angle\gamma}\left( \hat{\theta} \right)}.}}$

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a wireless system according to anembodiment.

FIGS. 2A-C illustrate observation windows for OFDM symbols withdifferent time offsets.

FIG. 3 is a flowchart describing a joint time offset and carrierfrequency offset maximum likelihood (ML) estimation operation accordingto an embodiment.

FIG. 4 is a plot showing the mean square error performance of two typesof joint ML estimators as a function of time offset.

FIG. 5 is a plot showing the mean square error performance of the twojoint ML estimators for varying signal-to-noise ration (SNR).

DETAILED DESCRIPTION

FIG. 1 shows a wireless communication system 100 according to anembodiment. The wireless communication system includes a transmitter 102and a receiver 104 that communicate over a wireless channel 106. Thetransmitter 102 and receiver 104 may be implemented in two differenttransceivers, each transceiver including both a transmitter section anda receiver section.

The wireless communication system 100 may be implemented in a wirelesslocal Area Network (WLAN) that complies with the IEEE 802.11 standards(including IEEE 802.11, 802.11a, 802.11b, 802.11g, and 802.11n). TheIEEE 802.11 standards describe OFDM systems and the protocols used bysuch systems. In an OFDM system, a data stream is split into multiplesubstreams, each of which is sent over a different subcarrier frequency(also referred to as a “tone”). For example, in IEEE 802.11a systems,OFDM symbols include 64 tones (with 48 active data tones) indexed as{−32, −31, . . . , −1, 0, 1, . . . , 30, 31}, where 0 is the DC toneindex. The DC tone is not used to transmit information.

At the transmitter 102, N complex data symbols are transformed totime-domain samples by an inverse discrete Fourier transform (IDFT)module 108. A cyclic prefix may be added to the body of the OFDM symbolto avoid interference (ISI) and preserve orthogonality betweensubcarriers. The cyclic prefix may include copies of the last N_(g)samples of the N time-domain samples. The cyclic prefix is appended as apreamble to the N time-domain samples to form the complete OFDM symbolwith N_(t)=N_(g)+N samples.

The OFDM symbols are converted to a single data stream by aparallel-to-serial (P/S) converter 110 and concatenated serially. Thediscrete symbols are converted to analog signals by a digital-to-analogconverter (DAC) 112 and lowpass filtered for radio frequency (RF)upconversion by an RF module 114. The OFDM symbols are transmitted overthe wireless channel 106 to the receiver 104, which performs the inverseprocess of the transmitter 102.

At the receiver 104, the received signals are down converted andfiltered by an RF module 120 and converted to a digital data stream byan analog-to-digital converter (ADC) 122. A joint frequency offset andtime offset estimator 124 may be used for frequency and symbolsynchronization. The estimator 124 may include a receiver 150 to receivesamples from the ADC 122, a framer 152 to observe a window of samples ofthe received OFDM symbols, and a calculator 154 to estimate the carrierfrequency offset ε and time offset θ. The estimated frequency offset{circumflex over (ε)} may be fed back to the downconverter 120 tocorrect carrier frequency and the estimated time offset {circumflex over(θ)} may be fed to an FFT window alignment module 126 to enable thealignment module to determine the boundary of the OFDM symbols forproper FFT demodulation. The estimated frequency offset can be used tocorrect the carrier frequency in the digital domain. The data stream isthen converted into parallel substreams by a serial-to-parallel (S/P)converter 128 and transformed into N tones by a DFT module 130.

For the additive white Gaussian noise (AWGN) channel, the receivedsignal y[n] with time offset θ and frequency offset ε can expressed asy[n]=x[n−θ]e ^(j2πεn/N) +z[n],  (1)

where x[n] and z[n] are the transmit signal with variance σ_(x) ² andwhite Gaussian noise with variance σ_(z) ², respectively. When N islarge, the transmit signal can be regarded as a Gaussian process. Sincethe transmit signal is independent from the noise, the received signalcan be modeled as a Gaussian process with the following autocorrelationproperties:

$\begin{matrix}{{E\left\lbrack {y\left\{ n \right\rbrack{y^{*}\left\lbrack {n + m} \right\rbrack}} \right\rbrack} = \left\{ {\begin{matrix}{{{\left( {\sigma_{x}^{2} + \sigma_{z}^{2}} \right){\delta\lbrack m\rbrack}} + {\sigma_{x}^{2}{\mathbb{e}}^{- {j2\pi ɛ}}{\delta\left\lbrack {m - N} \right\rbrack}}},} & {m \in A_{\theta}} \\{{{\left( {\sigma_{x}^{2} + \sigma_{z}^{2}} \right){\delta\lbrack m\rbrack}} + {\sigma_{x}^{2}{\mathbb{e}}^{j2\pi ɛ}{\delta\left\lbrack {m + N} \right\rbrack}}},} & {m \in B_{\theta}} \\{{\left( {\sigma_{x}^{2} + \sigma_{z}^{2}} \right){\delta\lbrack m\rbrack}},} & {otherwise}\end{matrix},} \right.} & (2) \\{\mspace{20mu}{{\delta\lbrack m\rbrack} = \left\{ {\begin{matrix}1 & {m = 0} \\0 & {otherwise}\end{matrix},} \right.}} & (3) \\{\mspace{20mu}{{A_{k} = {\underset{i = {- \infty}}{\bigcup\limits^{\infty}}\left\{ {{k + {iN}_{t}},\ldots\mspace{14mu},{\left( {N_{g} - 1} \right) + k + {iN}_{t}}} \right\}}},{and}}} & (4) \\{\mspace{20mu}{{B_{k} = {\underset{i = {- \infty}}{\bigcup\limits^{\infty}}\left\{ {{N + k + {iN}_{t}},\ldots\mspace{14mu},{\left( {N_{t} - 1} \right) + k + {iN}_{t}}} \right\}}},}} & (5)\end{matrix}$

where N_(t)=N_(g)+N, as described above.

The sets A_(θ) and B_(θ) contain the indices of the cyclic prefixsamples and the useful samples that are duplicated in the cyclic prefix,respectively.

The estimator 124 may be a maximum likelihood (ML) estimator. MLestimators may be “blind” estimators, i.e., they do not require trainingsymbols or pilots, but rather exploit the fact that the cyclic prefixsamples are duplicates of part of the useful data samples in the OFDMsymbol. As is shown in FIGS. 2A-C, for an observation window 202 of2N_(t)−1 samples, depending on the time offset θ, the observation windowcontains a varying number of cyclic prefix samples whose correlateduseful samples are also inside the observation window.

In an embodiment, the estimator 124 may exploit all correlated samplesin the observation window. Also, the joint ML estimator may not belimited to an observation window of 2N_(t)−1, but may be performed for arange of observation window sizes, trading off precision for largerobservation windows with estimation speed for smaller observationwindows.

FIG. 3 is a flowchart describing a joint time offset and frequencyoffset ML estimation operation according to an embodiment. A window sizeM may be selected (block 301). The estimator 124 receives OFDM symbolsfrom the ADC 122 (block 302) and observes samples in a window having adefined size (block 304).

The estimator 124 calculates a correction value based on the number ofcorrelated cyclic prefix samples and useful samples in the window (block305), described below in Eq. (11), and performs an ML estimation usingsamples in the observation window and the correction factor (block 306).

For a given observation window size M, the received samples with indexnεA_(θ)∩I (where I={0, . . . , M−N−1}) are the only cyclic prefixsamples whose correlated useful samples are also inside the observationwindow. The estimator then uses the time offset estimation value tocalculate the frequency offset estimation value (block 308).

The time offset estimation and frequency offset estimation are given inEqs. (7) and (8) below and are derived as follows. For ML estimation,one should choose the time offset estimate m and frequency offset λ thatmaximize

$\begin{matrix}{{{\log\left( {{{f\left( {{y\lbrack 0\rbrack},\ldots\mspace{14mu},{y\left\lbrack {M - 1} \right\rbrack}} \right.}m},\lambda} \right)} = {{\log\left( {\prod\limits_{n \in {A_{m}\bigcap I}}\;{{f\left( {{y\lbrack n\rbrack},{{y\left\lbrack {n + N} \right\rbrack}❘m},\lambda} \right)}{\prod\limits_{n \notin {{({A_{m}\bigcup B_{m}})}\bigcap I}}{f\left( {{{y\lbrack n\rbrack}❘m},\lambda} \right)}}}} \right)} = {{\log\left( {\prod\limits_{n \in {A_{m}\bigcap I}}{\frac{f\left( {{y\lbrack n\rbrack},{{y\left\lbrack {n + N} \right\rbrack}❘m},\lambda} \right)}{{f\left( {{{y\lbrack n\rbrack}❘m},\lambda} \right)}{f\left( {{{y\left\lbrack {n + N} \right\rbrack}❘m},\lambda} \right)}}{\prod\limits_{n}\;{f\left( {{{y\lbrack n\rbrack}❘m},\lambda} \right)}}}} \right)} = {{\sum\limits_{n \in {A_{m}\bigcap I}}{\log\left( \frac{f\left( {{y\lbrack n\rbrack},{{y\left\lbrack {n + N} \right\rbrack}❘m},\lambda} \right)}{{f\left( {{{y\lbrack n\rbrack}❘m},\lambda} \right)}{f\left( {{{y\left\lbrack {n + N} \right\rbrack}❘m},\lambda} \right)}} \right)}} + C}}}},} & (6)\end{matrix}$

where C is a constant that does not depend on m and λ, and f(·|m,λ)denotes the conditional probability density function (pdf) of thevariables in its argument given m and λ. The conditional pdf f(·|m,λ)can be derived from Eq. (2).

By inserting the conditional pdf in Eq. (6) and manipulating Eq. (6)algebraically, the following time and frequency offset estimators can beobtained:

$\begin{matrix}{{\hat{\theta} = \;{\underset{0 \leq m \leq {N_{t} - 1}}{\arg\;\max}\left\{ {{{\gamma(m)}} - {{\rho\Phi}(m)} - {\Psi(m)}} \right\}}},} & (7) \\{{\hat{ɛ} = {{- \frac{1}{2\pi}}{{\angle\gamma}\left( \hat{\theta} \right)}}},} & (8)\end{matrix}$

where

$\begin{matrix}{{{\gamma(m)} = {\sum\limits_{n \in {A_{m}\bigcap I}}\;{{y\lbrack n\rbrack}{y^{*}\left\lbrack {n + N} \right\rbrack}}}},} & (9) \\{{{\Phi(m)} = {\frac{1}{2}{\sum\limits_{n \in {A_{m}\bigcap I}}\left( {{{y\lbrack n\rbrack}}^{2} + {{y\left\lbrack {n + N} \right\rbrack}}^{2}} \right)}}},} & (10) \\{{{\Psi(m)} = {{{A_{m}\bigcap I}}\frac{2\rho}{\left( {\sigma_{x}^{2} + \sigma_{z}^{2}} \right)\left( {1 - \rho^{2}} \right)}{\log\left( {1 - \rho^{2}} \right)}}},} & (11) \\{{\rho = \frac{\sigma_{x}^{2}}{\sigma_{x}^{2} + \sigma_{Z}^{2}}},} & (12)\end{matrix}$

where |A_(m)∩I| is the number of elements in the set A_(m)∩I.

The term Ψ(m) is used as a correction factor to account for the factthat for a given observation window, the number of cyclic prefix sampleswith correlated useful samples in the window depends on the position ofthe cyclic prefix samples in the observation window, except for thespecial case of N_(t)+N, where there are always N_(g) correlatedsamples.

It can be shown that ρ and Ψ(m) converge to one and zero, respectively,as the SNR increases. Accordingly, for very high SNR, the ML estimatormay be simplified by taking the autocorrelation (Eq. (9)) of thereceived signal for 0≦m≦N_(t)−1 and normalizing it by subtracting theenergy given in Eq. (10).

FIGS. 4 and 5 show the performance of a joint ML estimator according toan embodiment compared to a suboptimal joint ML estimator for anobservation window size M of 2N_(t)−1 samples. For the suboptimal MLestimator, it is assumed that a receiver observes N_(t)+N consecutivesamples. For the suboptimal estimator, the time offset estimate{circumflex over (θ)} and frequency offset estimate {circumflex over(ε)} are given by

$\begin{matrix}{{\hat{\theta} = {\underset{0 \leq m \leq {N_{t} - 1}}{\arg\;\max}\left\{ {{{\gamma(m)}} - {{\rho\Phi}(m)}} \right\}}},} & (13) \\{{\hat{ɛ} = {{- \frac{1}{2\pi}}{{\angle\gamma}\left( \hat{\theta} \right)}}},} & (14) \\{where} & \; \\{{{\gamma(m)} = {\underset{n = m}{\sum\limits^{m + N_{g} - 1}}\;{{y\lbrack n\rbrack}{y^{*}\left\lbrack {n + N} \right\rbrack}}}},{and}} & (15) \\{{{\Phi(m)} = {\frac{1}{2}{\underset{n = m}{\sum\limits^{m + N_{g} - 1}}\left( {{{y\lbrack n\rbrack}}^{2} + {{y\left\lbrack {n + N} \right\rbrack}}^{2}} \right)}}},} & (16)\end{matrix}$

where ρ is given above in Eq. (12).

Although this estimator was derived with the assumption that a receiverobserves only N_(t)+N samples, it requires the observation of 2N_(t)−1consecutive samples for proper operation, as can be seen from Eqs. (13),(15), and (16).

FIG. 4 shows the mean-square error performance 402, 404 as a function ofthe time offset θ for an ML estimator using the time offset estimator inEq. (7) and the suboptimal ML estimator, respectively. In this example,N_(g)=16 and N=64. As can be seen from the figure, the use of theoptimal ML estimator results in smaller mean-square error when the timeoffset is within the interval [0,N_(g)−1] and [N+1,N_(t)]. This occursbecause the observation window contains more than N_(g) correlatedvalues for the time offset in [0,N_(g)−1] and [N+1,N_(t)]. Theperformance of the frequency offset estimator exhibits similar trends.

FIG. 5 shows the mean-square error performance 502, 504 of an MLestimator using Eq. (7) and the suboptimal ML estimator, respectively,for varying SNR assuming that the time offset is uniformly distributedover [0,N_(t)−1]. As can be seen from the plot, the ML estimator usingEq. (7) requires approximately 1 dB to 2 dB less SNR than the suboptimalML estimator to achieve the same performance.

A number of embodiments have been described. Nevertheless, it will beunderstood that various modifications may be made without departing fromthe spirit and scope of the invention. For example, blocks in theflowchart may be skipped or performed out of order and still producedesirable results. Accordingly, other embodiments are within the scopeof the following claims.

1. A method comprising: receiving a signal comprising at least onesymbol, the symbol including a plurality of body samples and a pluralityof cyclic prefix samples; observing a plurality of samples in a window;calculating a first correction value based on a number of correlatedcyclic prefix samples that start at a first position within the windowand body samples in the window; calculating a second correction valuebased on a number of correlated cyclic prefix samples that start at asecond position within the window and body samples in the window,wherein the second position is different from the first position; andcalculating an estimated time offset value using samples in the windowand a plurality of correction values comprising the first correctionvalue and the second correction value.
 2. The method of claim 1, whereinsaid calculating the estimated time offset comprises performing amaximum likelihood estimation operation.
 3. The method of claim 1,wherein said receiving the signal comprises receiving a plurality ofOrthogonal Frequency Division Multiplexing (OFDM) symbols.
 4. The methodof claim 1, further comprising selecting a number of samples in thewindow, said number of samples corresponding to an accuracy of theestimated time offset value.
 5. The method of claim 1, furthercomprising using the estimated time offset value to calculate anestimated frequency offset value.
 6. The method of claim 1, wherein saidcalculating the estimated time offset value {circumflex over (θ)}comprises solving the equation:${\hat{\theta} = \;{\underset{0 \leq m \leq {N_{t} - 1}}{\arg\;\max}\left\{ {{{\gamma(m)}} - {{\rho\Phi}(m)} - {\Psi(m)}} \right\}}},$where $\begin{matrix}{{{\gamma(m)} = {\sum\limits_{n \in {A_{m}\bigcap I}}\;{{y\lbrack n\rbrack}{y^{*}\left\lbrack {n + N} \right\rbrack}}}},} \\{{{\Phi(m)} = {\frac{1}{2}{\sum\limits_{n \in {A_{m}\bigcap I}}\left( {{{y\lbrack n\rbrack}}^{2} + {{y\left\lbrack {n + N} \right\rbrack}}^{2}} \right)}}},} \\{{{\Psi(m)} = {{{A_{m}\bigcap I}}\frac{2\rho}{\left( {\sigma_{x}^{2} + \sigma_{z}^{2}} \right)\left( {1 - \rho^{2}} \right)}{\log\left( {1 - \rho^{2}} \right)}}},} \\{{\rho = \frac{\sigma_{x}^{2}}{\sigma_{x}^{2} + \sigma_{z}^{2}}},} \\{{A_{m} = {\underset{i = {- \infty}}{\bigcup\limits^{\infty}}\left\{ {{m + {iN}_{t}},\ldots\mspace{14mu},{\left( {N_{g} - 1} \right) + m + {iN}_{t}}} \right\}}},}\end{matrix}$ I={0, . . . , M−N−1}, where y[n] is a received sample withindex n, N is the number of samples in the body of a symbol, N_(g) isthe number of samples in the cyclic prefix of a symbol, N_(t) is thetotal number of samples in a symbol, M is the number of samples in thewindow, σ_(x) ² is the variance of a transmit signal, σ_(z) ² is thevariance of the white Gaussian noise, and Ψ(m) is a correction value. 7.The method of claim 6, further comprising: calculating an estimatedfrequency offset value {circumflex over (ε)} by solving the followingequation:$\hat{ɛ} = {{- \frac{1}{2\pi}}{{{\angle\gamma}\left( \hat{\theta} \right)}.}}$8. An apparatus comprising: an estimator comprising: a receiver toreceive a signal comprising at least one symbol, the symbol including aplurality of body samples and a plurality cyclic prefix samples, a framecircuit to observe a plurality of samples in a window; and a calculatorto calculate: a first correction value based on a number of correlatedcyclic prefix samples that start at a first position within the windowand body samples in the window, a second correction value based on anumber of correlated cyclic prefix samples that start at a secondposition within the window and body samples in the window, wherein thesecond position is different from the first position, and an estimatedtime offset value using samples in the window and a plurality ofcorrection values comprising the first correction value and the secondcorrection value.
 9. The apparatus of claim 8, wherein said calculatorperforms a maximum likelihood estimation operation to calculate theestimated time offset value.
 10. The apparatus of claim 8, wherein saidat least one symbol comprises an Orthogonal Frequency DivisionMultiplexing (OFDM) symbol.
 11. The apparatus of claim 8, wherein thenumber of samples in the window is selected based on a desired accuracyof the estimator.
 12. The apparatus of claim 8, wherein said calculatorfurther calculates an estimated frequency offset value based on theestimated time offset value.
 13. The apparatus of claim 8, wherein saidcalculator calculating the estimated time offset value {circumflex over(θ)} comprises solving the equation:${\hat{\theta} = {\underset{0 \leq m \leq {N_{t} - 1}}{\arg\max}\left\{ {{{\gamma(m)}} - {{\rho\Phi}(m)} - {\Psi(m)}} \right\}}},$where${{\gamma(m)} = {\sum\limits_{n \in {A_{m}\bigcap I}}\;{{y\lbrack n\rbrack}{y^{*}\left\lbrack {n + N} \right\rbrack}}}},{{\Phi(m)} = {\frac{1}{2}{\sum\limits_{n \in {A_{m}\bigcap I}}\left( {{{y\lbrack n\rbrack}}^{2} + {{y\left\lbrack {n + N} \right\rbrack}}^{2}} \right)}}},{{\Psi(m)} - {{{A_{m}\bigcap I}}\frac{2\rho}{\left( {\sigma_{x}^{2} + \sigma_{z}^{2}} \right)\left( {1 - \rho^{2}} \right)}{\log\left( {1 - \rho^{2}} \right)}}},{\rho = \frac{\sigma_{x}^{2}}{\sigma_{x}^{2} + \sigma_{z}^{2}}},{A_{m} = {\underset{i = \infty}{\bigcup\limits^{\infty}}\left\{ {{m + {iN}_{t}},\ldots\mspace{14mu},{\left( {N_{g} - 1} \right) + m + {iN}_{t}}} \right\}}},$I={0, . . . , M−N−1}, where y[n] is a received sample with index n, N isthe number of samples in the body of a symbol, N_(g) is the number ofsamples in the cyclic prefix of a symbol, N_(t) is the total number ofsamples in a symbol, M is the number of samples in the window, σ_(x) ²is the variance of a transmit signal, σ_(z) ^(i) is the variance of thewhite Gaussian noise, and Ψ(m) is a correction value.
 14. The apparatusof claim 13, wherein said calculator calculates an estimated frequencyoffset value {circumflex over (ε)} by solving the following equation:$\hat{ɛ} = {{- \frac{1}{2\pi}}{{{\angle\gamma}\left( \hat{\theta} \right)}.}}$15. An apparatus comprising: estimator means comprising: means forreceiving a signal comprising at least one symbol, the symbol includinga plurality of body samples and a plurality cyclic prefix samples, meansfor observing a plurality of samples in a window; means for calculatinga first correction value based on a number of correlated cyclic prefixsamples that start at a first position within the window and bodysamples in the window; means for calculating a second correction valuebased on a number of correlated cyclic prefix samples that start at asecond position within the window and body samples in the window whereinthe second position is different from the first position; and means forcalculating an estimated time offset value using samples in the windowand a plurality of correction values comprising the first correctionvalue and the second correction value.
 16. The apparatus of claim 15,further comprising means for performing a maximum likelihood estimationoperation.
 17. The apparatus of claim 15, further comprising means forreceiving a plurality of Orthogonal Frequency Division Multiplexing(OFDM) symbols.
 18. The apparatus of claim 15, wherein the number ofsamples in the window is selected based on a desired accuracy of theestimated time offset value.
 19. The apparatus of claim 15, furthercomprising means for calculating an estimated frequency offset valueusing the estimated time offset value.
 20. The apparatus of claim 15,further comprising means for solving the equation:${\hat{\theta} - {\arg\;{\max\limits_{0 \leq m \leq {N_{t} - 1}}\left\{ {{{\gamma(m)}} - {{\rho\Phi}(m)} - {\Psi(m)}} \right\}}}},$where${{\gamma(m)} = {\sum\limits_{n \in {A_{m}\bigcap I}}\;{{y\lbrack n\rbrack}{y^{*}\left\lbrack {n + N} \right\rbrack}}}},{{\Phi(m)} = {\frac{1}{2}{\sum\limits_{n \in {A_{m}\bigcap I}}\left( {{{y\lbrack n\rbrack}}^{2} + {{y\left\lbrack {n + N} \right\rbrack}}^{2}} \right)}}},{{\Psi(m)} - {{{A_{m}\bigcap I}}\frac{2\rho}{\left( {\sigma_{x}^{2} + \sigma_{z}^{2}} \right)\left( {1 - \rho^{2}} \right)}{\log\left( {1 - \rho^{2}} \right)}}},{\rho = \frac{\sigma_{x}^{2}}{\sigma_{x}^{2} + \sigma_{z}^{2}}},{A_{m} = {\underset{i = \infty}{\bigcup\limits^{\infty}}\left\{ {{m + {iN}_{t}},\ldots\mspace{14mu},{\left( {N_{g} - 1} \right) + m + {iN}_{t}}} \right\}}},$I={0, . . . , M−N−1}, where y[n] is a received sample with index n, N isthe number of samples in the body of a symbol, N_(g) is the number ofsamples in the cyclic prefix of a symbol, N_(t) is the total number ofsamples in a symbol, M is the number of samples in the window, σ_(x) ²is the variance of a transmit signal, σ_(z) ² is the variance of thewhite Gaussian noise, and Ψ(m) is a correction value.
 21. The apparatusof claim 20, further comprising: means for calculating an estimatedfrequency offset value {circumflex over (ε)} by solving the followingequation:$\hat{ɛ} = {{- \frac{1}{2\pi}}{{{\angle\gamma}\left( \hat{\theta} \right)}.}}$22. A computer program product, encoded on a computer-readable medium,operable to cause data processing apparatus to perform operationscomprising: receiving a signal comprising at least one symbol, thesymbol including a plurality of body samples and a plurality of cyclicprefix samples; observing a plurality of samples in a window;calculating a first correction value based on a number of correlatedcyclic prefix samples that start at a first position within the windowand body samples in the window; calculating a second correction valuebased on a number of correlated cyclic prefix samples that start at asecond position within the window and body samples in the window,wherein the second position is different from the first position; andcalculating an estimated time offset value using samples in the windowand a plurality of correction values comprising the first correctionvalue and the second correction value.
 23. The computer program productof claim 22, wherein said calculating the estimated time offsetcomprises performing a maximum likelihood estimation operation.
 24. Thecomputer program product of claim 22, wherein said receiving a signalcomprises receiving a plurality of Orthogonal Frequency DivisionMultiplexing (OFDM) symbols.
 25. The computer program product of claim22, further comprising selecting a number of samples in the window, saidnumber of samples corresponding to an accuracy of the estimated timeoffset value.
 26. The computer program product of claim 22, furthercomprising using the estimated time offset value to calculate anestimated frequency offset value.
 27. The computer program product ofclaim 22, wherein said calculating the estimated time offset value{circumflex over (θ)} comprises solving the equation:${\hat{\theta} - {\arg\;{\max\limits_{0 \leq m \leq {N_{t} - 1}}\left\{ {{{\gamma(m)}} - {{\rho\Phi}(m)} - {\Psi(m)}} \right\}}}},$where${{\gamma(m)} = {\sum\limits_{n \in {A_{m}\bigcap I}}\;{{y\lbrack n\rbrack}{y^{*}\left\lbrack {n + N} \right\rbrack}}}},{{\Phi(m)} = {\frac{1}{2}{\sum\limits_{n \in {A_{m}\bigcap I}}\left( {{{y\lbrack n\rbrack}}^{2} + {{y\left\lbrack {n + N} \right\rbrack}}^{2}} \right)}}},{{\Psi(m)} - {{{A_{m}\bigcap I}}\frac{2\rho}{\left( {\sigma_{x}^{2} + \sigma_{z}^{2}} \right)\left( {1 - \rho^{2}} \right)}{\log\left( {1 - \rho^{2}} \right)}}},{\rho = \frac{\sigma_{x}^{2}}{\sigma_{x}^{2} + \sigma_{z}^{2}}},{A_{m} = {\underset{i = \infty}{\bigcup\limits^{\infty}}\left\{ {{m + {iN}_{t}},\ldots\mspace{14mu},{\left( {N_{g} - 1} \right) + m + {iN}_{t}}} \right\}}},$I={0, . . . , M−N−1}, where y[n] is a received sample with index n, N isthe number of samples in the body of a symbol, N_(g) is the number ofsamples in the cyclic prefix of a symbol, N_(t) is the total number ofsamples in a symbol, M is the number of samples in the window, σ_(x) ²is the variance of a transmit signal, σ_(z) ² is the variance of thewhite Gaussian noise, and Ψ(m) is a correction value.
 28. The computerprogram product of claim 27, further comprising: calculating anestimated frequency offset value {circumflex over (ε)} by solving thefollowing equation:$\hat{ɛ} = {{- \frac{1}{2\pi}}{{{\angle\gamma}\left( \hat{\theta} \right)}.}}$29. A system comprising: one or more antennas to receive signals from achannel; and an estimator comprising: a receiver to receive at least onesymbol from the one or more antennas, the symbol including a pluralityof body samples and a plurality cyclic prefix samples, a frame circuitto observe a plurality of samples in a window; and a calculator tocalculate: a first correction value based on a number of correlatedcyclic prefix samples that start at a first position within the windowand body samples in the window, a second correction value based on anumber of correlated cyclic prefix samples that start at a secondposition within the window and body samples in the window, wherein thesecond position is different from the first position, and an estimatedtime offset value using samples in the window and a plurality ofcorrection values comprising the first correction value and the secondcorrection value.
 30. The system of claim 29, wherein said calculatingthe estimated time offset comprises performing a maximum likelihoodestimation operation.
 31. The system of claim 29, wherein said at leastone symbol comprises an Orthogonal Frequency Division Multiplexing(OFDM) symbol.
 32. The system of claim 29, wherein the number of samplesin the window is selected based on a desired accuracy of the estimator.33. The system of claim 29, wherein said calculator further calculatesan estimated frequency offset value based on the estimated time offsetvalue.
 34. The system of claim 29, wherein said calculator calculatingthe estimated time offset value {circumflex over (θ)} comprises solvingthe equation:${\hat{\theta} - {\arg\;{\max\limits_{0 \leq m \leq {N_{t} - 1}}\left\{ {{{\gamma(m)}} - {{\rho\Phi}(m)} - {\Psi(m)}} \right\}}}},$where${{\gamma(m)} = {\sum\limits_{n \in {A_{m}\bigcap I}}\;{{y\lbrack n\rbrack}{y^{*}\left\lbrack {n + N} \right\rbrack}}}},{{\Phi(m)} = {\frac{1}{2}{\sum\limits_{n \in {A_{m}\bigcap I}}\left( {{{y\lbrack n\rbrack}}^{2} + {{y\left\lbrack {n + N} \right\rbrack}}^{2}} \right)}}},{{\Psi(m)} - {{{A_{m}\bigcap I}}\frac{2\rho}{\left( {\sigma_{x}^{2} + \sigma_{z}^{2}} \right)\left( {1 - \rho^{2}} \right)}{\log\left( {1 - \rho^{2}} \right)}}},{\rho = \frac{\sigma_{x}^{2}}{\sigma_{x}^{2} + \sigma_{z}^{2}}},{A_{m} = {\underset{i = \infty}{\bigcup\limits^{\infty}}\left\{ {{m + {iN}_{t}},\ldots\mspace{14mu},{\left( {N_{g} - 1} \right) + m + {iN}_{t}}} \right\}}},$I={0, . . . , M−N−1}, where y[n] is a received sample with index n, N isthe number of samples in the body of a symbol, N_(g) is the number ofsamples in the cyclic prefix of a symbol, N_(t) is the total number ofsamples in a symbol, M is the number of samples in the window, σ_(x) ²is the variance of a transmit signal, σ_(z) ² is the variance of thewhite Gaussian noise, and Ψ(m) is a correction value.
 35. The system ofclaim 34, wherein said calculator calculates an estimated frequencyoffset value {circumflex over (ε)} by solving the following equation:$\hat{ɛ} = {{- \frac{1}{2\pi}}{{{\angle\gamma}\left( \hat{\theta} \right)}.}}$36. A system comprising: antenna means for receiving signals from achannel; and estimator means comprising: means for receiving a signalcomprising at least one symbol from the antenna means, the symbolincluding a plurality of body samples and a plurality cyclic prefixsamples, means for observing a plurality of samples in a window; meansfor calculating a first correction value based on a number of correlatedcyclic prefix samples that start at a first position within the windowand body samples in the window; means for calculating a secondcorrection value based on a number of correlated cyclic prefix samplesthat start at a second position within the window and body samples inthe window, wherein the second position is different from the firstposition; and means for calculating an estimated time offset value usingsamples in the window and a plurality of correction values comprisingthe first correction value and the second correction value.
 37. Thesystem of claim 36, further comprising means for performing a maximumlikelihood estimation operation.
 38. The system of claim 36, furthercomprising means for receiving a plurality of Orthogonal FrequencyDivision Multiplexing (OFDM) symbols.
 39. The system of claim 36,wherein the number of samples in the window is selected based on adesired accuracy of the estimator.
 40. The system of claim 36, furthercomprising means for calculating an estimated frequency offset valueusing the estimated time offset value.
 41. The system of claim 36,further comprising means for solving the equation:${\hat{\theta} - {\arg\;{\max\limits_{0 \leq m \leq {N_{t} - 1}}\left\{ {{{\gamma(m)}} - {{\rho\Phi}(m)} - {\Psi(m)}} \right\}}}},$where${{\gamma(m)} = {\sum\limits_{n \in {A_{m}\bigcap I}}\;{{y\lbrack n\rbrack}{y^{*}\left\lbrack {n + N} \right\rbrack}}}},{{\Phi(m)} = {\frac{1}{2}{\sum\limits_{n \in {A_{m}\bigcap I}}\left( {{{y\lbrack n\rbrack}}^{2} + {{y\left\lbrack {n + N} \right\rbrack}}^{2}} \right)}}},{{\Psi(m)} - {{{A_{m}\bigcap I}}\frac{2\rho}{\left( {\sigma_{x}^{2} + \sigma_{z}^{2}} \right)\left( {1 - \rho^{2}} \right)}{\log\left( {1 - \rho^{2}} \right)}}},{\rho = \frac{\sigma_{x}^{2}}{\sigma_{x}^{2} + \sigma_{z}^{2}}},{A_{m} = {\underset{i = \infty}{\bigcup\limits^{\infty}}\left\{ {{m + {iN}_{t}},\ldots\mspace{14mu},{\left( {N_{g} - 1} \right) + m + {iN}_{t}}} \right\}}},$I={0, . . . , M−N−1}, where y[n] is a received sample with index n, N isthe number of samples in the body of a symbol, N_(g) is the number ofsamples in the cyclic prefix of a symbol, N_(t) is the total number ofsamples in a symbol, M is the number of samples in the window, σ_(x) ²is the variance of a transmit signal, σ_(z) ² is the variance of thewhite Gaussian noise, and Ψ(m) is a correction value.
 42. The system ofclaim 41, further comprising: means for calculating an estimatedfrequency offset value {circumflex over (ε)} by solving the followingequation:$\hat{ɛ} = {{- \frac{1}{2\pi}}{{{\angle\gamma}\left( \hat{\theta} \right)}.}}$